Discrete part of the second Lagrange spectrum

Abstract

Given an irrational number $\alpha$ consider its irrationality measure function $\psi_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z}}\|q\alpha\|$. The set of all values of $\lambda(\alpha)=(\limsup\limits_{t\to\infty} t\psi_{\alpha}(t))^{-1}$ where $\alpha $ runs through the set $\mathbb{R}\setminus\mathbb{Q}$ is called the Lagrange spectrum $\mathbb{L}$. In a paper by Moshchevitin an irrationality measure function $\psi^{[2]}_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z},q\ne q_i}\|q\alpha\|$ was introduced. In other words, we consider the best approximations by fractions, whose denominators are not the denominators of the convergents to $\alpha$. Replacing the function $\psi_{\alpha}$ in the definition of $\mathbb{L}$ by $\psi^{[2]}_{\alpha}$, one can get a set $\mathbb{L}_2$ which is called the ''second'' Lagrange spectrum. In this paper we give the complete structure of discrete part of $\mathbb{L}_2$.